Papers

DomiRank is a node-importance algorithm for unweighted networks, defined by a dynamical model governed by a competition-strength parameter, a dominance threshold, and a decay rate. We establish three structural results. First, on symmetry. Automorphism-related vertices share the same DomiRank value, and a graph whose values are pairwise distinct must be asymmetric. The orbit count bounds the number of distinct values, and regular and vertex-transitive graphs admit closed forms. Second, on entropy. The maximum of the DomiRank entropy over connected graphs, log_2 N, is attained if and only if the graph is regular. The initial entropy derivative is the degree-log-degree edge covariance, making the decrease rigorous near vanishing competition for structurally disassortative graphs, and an assortative counterexample shows it does not extend to all graphs; a sufficient-condition theorem and a refined conjecture cover the rest. Third, on spectrum. The normalized distribution is sigma-invariant if and only if the degree vector is an adjacency-matrix eigenvector. The parameter sigma interpolates continuously between degree and smallest-eigenvector centrality within the resolvent (Bonacich-type) family, its sensitivity being a second-order suppression operator on the degree vector. Graph transformations are also characterized, including the m-product. Experiments on four real networks and synthetic families up to 10^4 vertices confirm the interpolation and exhibit monotone entropy decrease, continuous ranking interpolation, decaying positive fraction and ranking reversals; where the positive fraction has decayed far from one, the limit describes the solution vector rather than the induced ranking. DomiRank is thus a tunable complement to principal-eigenvector centrality. A companion study quantifies how much of its task-level edge over a degree baseline survives degree correction.

Let $A$ be an Artin algebra and let $T$ be a $\tau$-tilting right $A$-module. Put $C=A/{\rm Ann}_A(T)$. We prove that finite little finitistic dimension of $C^{\rm op}$, together with a finite range of self-extension vanishing for $T$, forces $T$ to be a classical $1$-tilting module. We also show that if the finitistic conjecture holds for all Artin algebras then the self-orthogonal $\tau$-tilting conjecture holds for all Artin algebras. Moreover, a counterexample to the self-orthogonal $\tau$-tilting conjecture will give a counterexample to the finitistic dimension conjecture. As an application, the self-orthogonal $\tau$-tilting conjecture holds for Artin algebras with radical cube zero.

We construct projective Calabi-Yau threefolds with isolated Gorenstein canonical singularities that admit no smoothing, as anticanonical hypersurfaces in Gorenstein Fano toric fourfolds. One example has a unique singularity, the anticanonical cone over the first Hirzebruch surface. Another has a unique singularity whose reduced miniversal base is a smooth curve but whose deformations are all singular. The construction combines Altmann's deformation theory with the smoothing criterion of Corti-Filip-Petracci: a two-dimensional face of the ambient fan polytope with primitive edges obstructs smoothing if it has no Minkowski decomposition into unit segments and unimodular triangles. We also classify the 217 isolated Gorenstein toric threefold singularities whose polygon edge vectors have coordinates in $[-2,2]$ in some lattice basis, recording their deformation types and smoothing components. Subject to completeness and absence of repetitions in the database copy used, a scan of the Kreuzer-Skarke classification finds this obstruction in $39{,}175{,}536$ of its $473{,}800{,}776$ polytopes (approximately $8.27\%$), whose generic anticanonical hypersurfaces therefore admit no smoothing.

Let $F$ be an effective balanced rational two-dimensional fan in a rank-four lattice. We introduce an affine spherical mass \[ M_*(F)=\inf_{covol_g(N)=1}M_g(F) \] and prove the two-sided comparison \[ \sqrt{\frac83}\,m(F)\leq M_*(F)\leq 3\sqrt{2\,q(F)}. \] Here $q(F)=deg(F\cdot F)$ is stable self-intersection and $m(F)$ is the least degree of $F$ under a primitive rank-two lattice quotient. For a surjection $A$, one has $deg(A_*F)=deg(F\cdot[\ker A_R])$; hence $m(F)$ is equivalently the least stable intersection degree with a complete weight-one rational two-plane. In particular, \[ q(F)\geq\frac4{27}m(F)^2. \] The proof combines an exact Gaussian–kinematic formula for mixed stable intersection with an affine normalization principle for stationary spherical links. In dimension four, the identification $Gr^+(2,4)\simeq S^2\times S^2$ converts ambient isotropy into a sharp second-moment transversality estimate. We prove affine covariance for $M_*$, derive equality and stability conditions for the transversality estimate, and give examples where ambient isotropy coexists with zero mixed intersection, as well as a three-fan in six-space with zero self-intersection. Determining the optimal numerical constant remains open.

We introduce a phase-angle framework for constructing and classifying n-tuple-periodic solutions of the incompressible Navier-Stokes equations, and use it to determine, dimension by dimension, when such solutions exist and why they fail. The velocity field is a difference of two n-fold cyclic trigonometric products, made divergence-free for every phase and every n >= 3 by a telescoping identity. The problem then reduces to two explicit conditions on the n phase angles: a reduction condition, necessary and algebraic, and a pressure-integrability condition, sufficient and differential. Separating them allows a failed construction to be located rather than merely recorded. Unlike the stream-function route on which earlier tri-periodic solutions rest, the framework is not confined to three dimensions, and does not presuppose the Beltrami property. For 3 <= n <= 8 we prove that unforced solutions exist at n=3 and n=4 and nowhere else. In three dimensions the admissible phases form exactly two families, mirror images distinguished by the sign of the helicity; these families are known, and we establish that the construction admits no others, settling a completeness question left open when they were found. In four dimensions integrability reduces a one-parameter curve to a single family, so the quarter-period offset is forced, not assumed. The obstructions divide by parity: arithmetic at n=5,7, where every admissible phase vector annihilates the field; differential at n=6,8, where the field survives but its convective term is not a gradient. Every obstructed dimension nevertheless carries an exact forced solution in closed form. These are not mere self-balancing: the phases are those at which strain and rotation are in exact pointwise balance, so that no pressure is generated at all. What obstructs the unforced problem becomes, in the forced problem, an exactly prescribable input.

We study profinite rigidity of real hyperbolic lattices, with the main application in dimension three. The basic tool is a profinite version of the dual cube complex of a pair of filling curves on a surface, which we call a profinite cubical residue complex. We prove that an equivariant isomorphism between the completions of two such complexes is induced, after conjugation by a single element, by an isomorphism of the discrete groups and complexes, and that the same holds for free cocompact cellular actions of finitely generated residually finite groups on connected, locally finite, finite dimensional regular CW complexes. On a closed orientable surface of genus at least two, a theory of divisible fillings determines the completed cut tree of a nonseparating simple closed curve. Combined with the homological intersection criterion of Boggi and Zalesskii, this determines the completed dual square complex of a nonseparating filling pair, and hence realizes every isomorphism of profinite surface groups which preserves the two marked cyclic subgroups. In dimension three we combine these results with virtual fibering, Liu's profinite invariants of hyperbolic $3$-manifolds and the goodness of $3$-manifold groups. Massey products and virtual domination give cohomological integrality, and pseudo-Anosov dynamics, periodic orbit traces and cross sections of suspension flows give a correspondence of prime periodic orbits together with the fiber curves needed for the realization. Profinite rigidity of closed and of cusped lattices follows, the cusped case through cyclic orbifold fillings, and we obtain $\Out(\Gamma)\cong\Out(\wh\Gamma)$ for every lattice of $\PSL_2(\C)$.

We show that the only degree which cannot co-enumerate a non-trivial $\Pi^0_1$-immune real is $\emptyset$, resolving a conjecture of the author. The main ingredient is that if a real $A$ has that all $A$-maximal sets are c.e., then $A$ has c.e. degree, which can be proved via a coding mechanism and two classical results of Lachlan.

Dynamical systems theory primarily concerns the study of different forms of invariance, such as invariant sets, densities, measures, and observables. Ergodic systems are a special class of dynamical systems which are measure theoretically irreducible. In spite of this specialized property they dominate most discussions on ergodic theory because of the ergodic decomposition theorem. The viewpoint that any dynamical system is a composite of multiple ergodic components enables us to partition the face space into the basins of the different measures. The coexistence and mutual connections between these various coexisting subsystems are illuminated very effectively using the language of Category theory (CT). Some recent advancements have shown how the essence of measure theoretic dynamical systems can be captured through the formalism of Markov categories. This formalism captures the essential features of invariance and ergodicity through the language of limits and colimits. This article continues that formalism by studying the concept of joins using the language of spans and push-outs. A classical result is re-proven which establishes the connection between joins, ergodicity and mixing. In the process, the categorical language is refined to capture the notion of sub sigma-algebras and their invariance.

For a partition $\lambda$, let $G_\lambda^{(\beta)}$ be the stable $\beta$-Grothendieck polynomial attached to $\lambda$. Each homogeneous component of the $\beta = 1$ specialization $G_\lambda^{(1)}$ is Schur-positive and hence positive in the fundamental basis of quasisymmetric functions. For $m\ge|\lambda|$, let $G_{\lambda,m}^{(1)}$ be the homogeneous degree $m$ component of $G_\lambda^{(1)}$. In this paper, we first give a direct proof of an expansion of $G_{\lambda,m}^{(1)}$ in the fundamental basis in terms of standard set-valued tableaux. We then use these tableaux as a basis to define a module of the $0$-Hecke algebra and show that the quasisymmetric characteristic of the resulting module is $G_{\lambda,m}^{(1)}$. We further show that this module decomposes as a direct sum of weak Bruhat interval modules.

Let $\Lambda$ be a compact Bohr–Sommerfeld Lagrangian submanifold of a compact Kähler manifold equipped with a holomorphic prequantum line bundle. We study the asymptotic expansion of the Lagrangian states associated with $\Lambda$. In particular, we compute explicitly the first nontrivial correction term and show that it is expressed in terms of geometric invariants of the ambient Kähler manifold and the Lagrangian submanifold, including their scalar curvatures, the second fundamental form, and the mean curvature. As a consequence, we obtain the corresponding second-order asymptotic formula for the $L^2$-norm of the Lagrangian states.

Considering the computational advantages of virtual element methods (VEM), this work employs a conforming VEM for the numerical approximation of the Kelvin–Voigt model of viscoelasticity. For clarity and simplicity, we focus on the primal formulation. The spatial discretization is carried out using the virtual element method, while the temporal discretization is handled via the {second-order Crank–Nicolson} scheme. We establish the well-posedness of both the semi-discrete and fully discrete problems and derive a priori error estimates. Several representative numerical examples are presented to validate the theoretical results and to demonstrate the effectiveness of the proposed formulation.

Hyperbolic conservation laws pose a challenging setting for variational data assimilation, since the Gaussian assumption imposes a smoothness that smears jump discontinuities. A sparsity promoting regularization can help to mitigate this problem by incorporating a structural prior into the variational objective. Care must be taken when choosing this prior because it serves to counterbalance both a structurally misspecified background-error covariance, which introduces spurious oscillations over smooth regions, as well as spatially sparse observations, which leave too little data to constrain the state. Moreover, state variables of hyperbolic conservation laws typically do not have piecewise constant structure, which is an assumption made when using standard sparsity-promoting operators, such as total variation. Indeed the underlying variability of hyperbolic conservation law solutions are neither known in advance nor fixed. Hence using higher order total variation is also not suitable. Here we introduce a new regularization term for data assimilation, built on the sparsity promoting residual transform that neither commits to a fixed smoothness order nor requires the underlying variability to be known a priori. We solve the resulting nonconvex objective within a three-dimensional variational framework using generalized sparse Bayesian learning. This approach replaces a global \(\ell _{1}\) penalty with a hierarchical conditional Gaussian prior on the residual transform, where a learned hyper-prior yields location-specific weights attached to each spatial point. Numerical experiments across scalar, shallow-water, and Euler test problems show that this regularization term resolves discontinuities more accurately than total variation, and that its advantage grows as the background-error covariance becomes misspecified and the observations become sparser.

Model-assisted estimation uses prediction rules to improve the efficiency of estimators of finite population parameters while retaining design-based inference. Although flexible prediction methods have been considered, existing theoretical results are largely method-specific. We develop a learner-agnostic framework that replaces separate analyses for individual learners with general conditions on the sampling design and prediction error. We connect design-aware and design-agnostic cross-fitting and characterize the sampling designs under which they yield conditional independence across folds. Under suitable conditions, conditional weighting gives exact design-unbiasedness. We establish first-order equivalence to oracle estimators, leading to design consistency and asymptotic normality, and clarify when conditional and original inclusion probabilities yield the same first-order behavior. We propose consistent variance estimators based on cross-fitted residuals and construct asymptotically valid confidence intervals. Under additional model and regularity conditions, we establish asymptotic optimality through attainment of the Godambe–Joshi lower bound. Simulations show that cross-fitting substantially reduces finite-sample bias and improves variance estimation and coverage with adaptive learners.

For practical $N$ let $h(N)$ be the least $k$ such that every integer $1\le m\le N$ is a sum of at most $k$ distinct divisors of $N$. We prove $h(n!)\le(2\log2+o(1))\,n/\log n$. This improves the bounds of order $n/(\log n)^{1/2-\varepsilon}$ established in Tenenbaum-Yokota's Lemma 4 and Yokota's 1995 knapsack note. We combine their decreasing greedy construction with the sharper factorial divisor-gap estimate of Berend-Harmse. Counting the steps separately below and above $\sqrt{n!}$, with the upper range handled through reciprocal divisors, retains the leading coefficient in the gap exponent and yields the explicit constant $2\log2$.

Using estimates established in a previous paper, we prove quantitative universality results for cokernels of products of i.i.d. random matrices and for flags associated to k-tuples of random i.i.d. matrices, over finite local rings. In the case when the ring is a quotient of $\mathbb{Z}_p$, this gives a quantitative analogue of some previous results of Huang, Nguyen and Van Peski.

We consider the cokernel corners process for an i.i.d. matrix with entries in a finite local ring. When the distribution of the entries is uniform, this process is a Markov chain, and hence the ergodic theorem for Markov chains can be applied. This implies, in particular, that for uniformly distributed p-adic random matrices, the cokernels of the corners are distributed according to the Cohen-Lenstra measure, almost surely. The purpose of this note is to show that the conclusion of the ergodic theorem also holds for i.i.d matrices, provided that the distribution of the entries is not concentrated on the translate of a subring, or the translate of an ideal. This will follow from the bounds proved in a previous paper of the author.

We develop closed- and open-end procedures for monitoring changes in the marginal distribution of object-valued time series. The method combines two distance-based Hilbert-space embeddings, a monitoring-time-dependent projection, and self-normalization. It is computable entirely from pairwise distances, does not require long-run variance estimation, and admits exact recursive updates. Under weak temporal dependence, the closed-end null limit is a pivotal Brownian functional. For monitoring over an unbounded horizon, we introduce a growing monitoring-time weight, establish a maximal inequality and uniform remote-tail control, and derive an open-end pivotal limit together with an equivalent fixed-interval representation for critical-value simulation. Both procedures are consistent against fixed marginal changes and retain the $n^{-1/2}$ and $n^{-1/4}$ local detection boundaries associated with the linear and quadratic projection signals. Simulations with distribution-valued time series illustrate favorable numerical properties, and an application to monthly stock-return distributions illustrates the usefulness of our procedure. Further simulations with graph-valued time series and an application to spatial point processes of seismic activity are reported in the supplementary material.

We study the maximum-volume inscribed ellipsoid (MaxIE) problem for a polytope through a geometric iteration based on polarity. Given an interior point, the method forms the shifted polar polytope, computes its minimum-volume covering ellipsoid (MinCE), and then polarizes this covering ellipsoid back to obtain a new inscribed ellipsoid. This procedure was suggested by Khachiyan and Todd in 1993. Prior work studied basic polarity identities and gave an asymptotic-convergence argument for the exact process, but did not furnish a volume-ratio rate. Independently of that asymptotic argument, we develop a new analysis based on the log-volume of the polar MinCE as a potential function. We prove its convexity with an explicit gradient formula and establish global linear contraction of the potential gap along each trajectory. The contraction factor is existential and instance-dependent. This gives both a separate convergence proof and a finite volume-ratio iteration bound. We further analyze an inexact polarity process in which each MinCE subproblem is solved only approximately, and derive sufficient oracle tolerances for producing a prescribed volume approximation. Combining this outer analysis with existing algorithms for MinCE gives conditional arithmetic estimates based on the path-following Newton method, the barycentric coordinate descent, and the away-step Frank-Wolfe method. Numerical experiments compare these oracle choices with two MaxIE baselines and illustrate that performance depends on matching the MinCE solver to the instance geometry.

For every integer $n \geq 11$, we construct connected, irregular, nonisomorphic graphs on $n$ vertices with equal adjacency, Laplacian, signless Laplacian, normalized Laplacian, and Seidel spectra and equal independence, clique, chromatic, complement chromatic, and maximum-cut numbers, but distinct Lovász $\vartheta$-numbers. We join each graph in a fixed cospectral, nonisomorphic regular pair on ten vertices, due to van Dam and Haemers (2003), with $K_{n-10}$. We prove that these joins preserve equality of the five spectra and stated integer-valued invariants, and leave the respective Lovász numbers unchanged. We derive exact expressions for these numbers and prove that they are distinct. We also determine the cardinality-constrained maximum-cut profiles of both base graphs and their complements, yielding exact formulas and equality of maximum-cut numbers for the joins of either the base graphs or their complements with $K_{n-10}$. For $n=10$, we present a regular pair satisfying all stated properties except irregularity, followed by a connected, irregular, nonisomorphic pair satisfying all properties. An exhaustive computer search finds no connected, irregular, nonisomorphic pair on at most nine vertices sharing the first four spectra and all stated integer-valued invariants. Thus, ten is the smallest possible order, and such pairs exist for every $n \geq 10$. This extends and strengthens Sason's (2024) result for even $n \geq 14$, which used the neighbors splitting join operation and did not address Seidel matrices, complement chromatic numbers, or maximum-cut numbers of the graphs or joins of their complements. Thus, the Lovász number efficiently certifies nonisomorphism even when all five spectra and stated integer-valued invariants coincide.

Let $\Omega\subset\R^N$, $N\ge2$, be a bounded domain and let $1<p<\infty$. Inspired by the exponential case treated in \cite{BO2026}, we study the quasilinear Schrödinger–Maxwell system \[ \begin{cases} -\Delta_p u+\psi G'(u)=f &\text{in }\Omega,\\ -\Delta_p\psi=G(u) &\text{in }\Omega,\\ u=\psi=0 &\text{on }\partial\Omega, \end{cases} \] where $G\in C^1(\R)$ is even, convex, nonnegative and nontrivial, with $G(0)=0$, and satisfies \[ G(t)\lesssim 1+|G'(t)| \qquad\text{for large }|t|. \] Under the assumption \[ |f|\ln(1+|f|)\in L^1(\Omega), \] we prove the existence of a finite-energy weak solution and show that the solution is a saddle point of the associated functional.

The signatures of $p$-rough paths form a subgroup of sufficiently high-level truncated tensor algebras, whose inverse limit is a subgroup of the full tensor algebra. For $p \geq 1$, we provide a top-down description of the signature group as the inverse limit of finite-dimensional Carnot–Carathéodory geometries in the $p$-variation setting. We show that every compatible choice of metrics induces a topological tree structure on the inverse-limit group, under which the signature group is not a topological group. This extends the results of Enrico Le Donne and Roland Züst from bounded variation to rough paths. We also characterise the dependence of the inverse-limit groups and their metric completions on the choice of metric, identifying them with the tree-reduced path group of Horatio Boedihardjo, Xiang Geng, Terry Lyons, and Danyu Yang.

In this note, we provide a short and direct proof that approximately solving $Ax=b$ to relative error $\varepsilon$, where $A$ has condition number $\kappa$ and unrestricted dimension, requires $\Omega(\kappa\log(1/\varepsilon))$ matrix-vector multiplications in the worst case, even for randomized algorithms. This essentially recovers the lower bound of Dereziński, Epperly and Meyer [2026] for this setting, whose elegant and more general approach inspired us to seek a short direct proof. A straightforward reduction implies the classical $\Omega(\sqrt{\kappa}\log(1/\varepsilon))$ lower bound for optimizing strongly convex quadratic functions, applicable to randomized algorithms.

A modal logic is elementary if it is sound and complete with respect to an elementary class of Kripke frames. We prove that if a finitely axiomatizable modal logic $L$ is elementary, then the validity problem for $L$, which asks whether a given finite Kripke frame validates $L$, is in the circuit complexity class uniform $\mathsf{AC}^0$. As an application, we construct infinitely many finitely axiomatizable $\mathsf{K4}$-stable logics that are not elementary. This result provides a negative answer to Problem 6.4 in [Bezhanishvili et al., 2018] for $\mathsf{K4}$-stable logics.

Let $\mathfrak L$ be a Frobenius maximal parabolic subalgebra of $\mathfrak{sl}_n$. For any $F\in\mathfrak L^*$ for which the Kirillov form $B_F(x,y)=F([x,y])$ is non-degenerate, let $\widehat F$ denote the associated principal element. We prove that the multiplicities of the eigenvalues of $\operatorname{ad}_{\widehat F}$ on $\mathfrak L$ form a unimodal sequence symmetric about $\frac12$. We also prove that the multiplicities of the eigenvalues of $\operatorname{ad}_{\widehat F}$ on $\mathfrak{gl}_n$ form a unimodal sequence symmetric about $0$. The proof relates the ranked meander associated to $\mathfrak L$ to the order ideals of a related fence poset through Panyushev reduction. The known unimodality of the rank polynomial of the fence poset implies that of the meander, which in turn determines a good grading of $\mathfrak{gl}_n$ in the sense of Elashvili and Kac. We prove that this grading coincides with that induced by the principal element and that the pyramid associated to this grading may be filled in such a way that its good element $e$ lies in $\mathfrak L$. The two unimodality results then follow from the injectivity properties of $\operatorname{ad}_e$ coming from the good grading and the duality induced by the bilinear form $B_F$.

We provide evidence of a finite-time singularity in the 3D Euler equations on the unbounded domain. Using a physics-informed neural network (PINN) with a self-similar ansatz, we find an approximate singular profile for the Euler system at the critical blowup rate of $0.5$ and certify it using a spline representation. The transport field associated with the obtained profile has local outgoing property throughout the domain that suggests linear damping, a key stabilizing mechanism for the candidate profile. We also establish a framework for proving nonlinear stability of the approximate self-similar profile, reducing the analysis to a large but finite collection of explicit estimates and computable constants.

Reinforcement learning (RL) agents deployed in real-world environments are often vulnerable to adversarial perturbations in state observations, creating risks in safety-critical applications. Certification methods can improve robustness against adversarial perturbations by providing lower bounds on expected cumulative rewards. Existing certification methods, however, mainly focus on risk-neutral objectives. In this paper, we extend certification methods to risk-sensitive objectives by establishing lower bounds on the exponential utility of cumulative rewards under $l_{p}$-norm-bounded state adversarial perturbations ($1\leq p <\infty$). By introducing a $\phi$-divergence relaxation of the perturbation set, we formulate the risk-sensitive certification problem as a convex optimization and derive its dual to obtain a tractable approximation of the certified lower bound. We further propose an empirical method that improves certified lower bounds by selecting the training risk-aversion parameter $\beta$ independently of the risk level used during evaluation. Experiments on both OpenAI Gym environments and a machine replacement problem show that, compared to risk-neutral training, risk-averse training generally yields policies with higher certified lower bounds, particularly under larger perturbation budgets. Moreover, under both risk-neutral and risk-averse evaluation settings, increasing risk aversion during training leads to non-monotonic certification performance, where certified lower bounds initially improve but eventually decrease due to overly conservative policies.

Constructing a confidence interval for the difference between two independent binomial proportions involves a nuisance direction that is not identified by the estimand. The one-sample Wilson score interval inverts a scalar score test, but has no direct bivariate analogue isolating the difference: inverting the joint normal approximation yields an elliptical region in the unit square, whereas the estimand \(p_1-p_2\) is one-dimensional. We define the Elliptically Optimal (EO) confidence interval as the range of \(p_1-p_2\) over this region and solve the resulting optimization problem in closed form, obtaining explicit bounds in six mutually exclusive and exhaustive cases. The solution admits a compact characterization: the EO interval is the score interval obtained by maximizing over the nuisance variance rather than estimating it. It is therefore the shortest interval obtained by projecting the elliptical region, and inherits its coverage guarantee. We derive the exact coverage excess, \(2[\Phi(z\mathcal{R})-\Phi(z)]\), where \(\mathcal{R}\) is the ratio of the least-favourable to the true standard deviation. The excess vanishes on an explicit line through the parameter space, is bounded by \(\alpha\), and is invariant under proportional scaling of the sample sizes. Exact enumeration of the binomial coverage shows that the Wald interval, whose variance estimator is downward biased by a factor \(1-1/n\) under balanced allocation, falls below nominal coverage almost everywhere. The EO interval never under-covers under the normal approximation and always yields admissible, non-degenerate bounds. Its price is over-coverage when both proportions are extreme, which we quantify exactly.

We develop a foundational distinction between well-definedness and predetermination. A persistent mathematical or computational object may return a unique answer whenever a query is made and preserve every previously returned answer, while its answers to as-yet unqueried inputs are not represented by one completed, history-independent extension. We call such an object well-defined but non-predetermined. We next formulate an observer-based semantics. All observers inhabit one shared computational world, their interactions modify one common global state, and they have access to one realized history. Under axioms of shared history, historical irreversibility, persistence, and historical indistinguishability, embedded observers cannot establish from the realized history alone whether its compatible extension was predetermined or progressively formed. A persistently evolutionary polynomial time computable oracle supplies a concrete example and yields a deterministic–nondeterministic separation under an explicitly history-sensitive semantics. We then derive a conditional result concerning the provability of \(P=NP\). The conclusion is an observer-relative unprovability theorem, not an unconditional proof of classical \(P\neq NP\).

In a recent numerical study, we found a high-precision singular profile for the Euler equations on the unbounded domain $\mathbb{R}^3$. The present manuscript complements that study by establishing in detail a preliminary framework for proving (nonlinear) stability of the approximate self-similar profile, reducing the analysis to a large but finite collection of explicit estimates and computable constants. Conditional on rigorous certification of the estimates and constants appearing in the argument, and on the candidate profile satisfying the required nonlinear stability conditions, the framework closes the stability proof and, crucially, allows the resulting stable rescaled profile to be reconstructed as an admissible solution in the original variables that becomes singular in finite physical time. With the overall stability and reconstruction mechanisms formulated, the remaining work within this approach is largely quantitative: determining whether the explicit constants and margins can be rigorously certified with sufficient positive margin and, where necessary, sharpening selected analytic estimates.

The AKSZ construction encodes the Batalin-Vilkovisky formulation of a topological field theory in terms of finite-dimensional geometric data. There are at least two ways to extend this approach to nontopological gauge theories. The first, due to Costello, replaces the spacetime exterior algebra with a more general DGCA that is not freely generated. The second replaces the symplectic structure with a degenerate presymplectic structure, which may also be non-regular. In this work, we show that both approaches are special cases of a more general algebraic AKSZ construction, in which the underlying algebras are allowed to be constrained and the presymplectic structure may be degenerate and non-regular. Within this framework, we explicitly relate Costello's formulation of the Chalmers-Siegel model to the presymplectic formulation, the latter admitting a first-principles derivation. We also construct Costello-like formulations of higher-form analogues of the Chalmers-Siegel model and of self-dual higher-spin gauge theories of Yang-Mills type, obtaining in the latter case a remarkably concise description. In so doing we propose a simple algebraic construction for the corresponding source space DGCAs underlying these examples.